Narrow Results

In this work, we study the stochastic resonance phenomenon in a bistable nonlinear dynamical system in the presence of an uncorrelated noise source whose distribution decays asymptotically as P(ξ) ∝ 1/ξ^2α. We investigate the influence of the decay exponent α on the transition rate and on the optimal noise intensity giving the maximum signal-to-noise ratio when a weak periodic signal is superposed to the external noise. We find that the transition rate achieves a maximum for a finite decay exponent α. However, the optimal noise intensity for stochastic resonance depicts a monotonic power-law correction relative to the usual behavior of nonlinear dynamical systems driven by Gaussian noises.

Unidirectional motion is achieved when a particle, moving under the influence of an underlying noise source, is subjected to a ratchet asymmetric periodic potential. Here, we investigate how deviations from the Gaussian nature of the noise distribution function impacts the average particle's current. The input noise is considered to be produced by a Langevin process including both multiplicative and additive random noise sources. The resulting input random signal has a power-law amplitude distribution and a finite correlation time. These features are controlled by the average of the multiplicative noise. We show that the average particle's velocity depends non-monotonically on the degree of non-Gaussianity of the input noise. It exhibits a maximum at an intermediate value of the effective power-law exponent that characterizes the asymptotic decay of the noise probability distribution function.

We present a general formalism for the dissipative dynamics of an arbitrary quantum system in the presence of a classical stochastic process. It is applicable to a wide range of physical situations, and in particular it can be used for qubit arrays in the presence of classical two-level systems (TLS). In this formalism, all decoherence rates appear as eigenvalues of an evolution matrix. Thus the method is linear, and the close analogy to Hamiltonian systems opens up a toolbox of well-developed methods such as perturbation theory and mean-field theory. We apply the method to the problem of a single qubit in the presence of TLS that give rise to pure dephasing 1/f noise and solve this problem exactly.

In this paper, the phenomenon of stochastic resonance in FitzHugh–Nagumo (FHN) neural system driven by correlated non-Gaussian noise and Gaussian white noise is investigated. First, the analytical expression of the stationary probability distribution is derived by using the path integral approach and the unified colored noise approximation. Then, we obtain the expression of signal-to-noise ratio (SNR) by applying the theory of two-state model. The results show that the phenomena of stochastic resonance and multiple stochastic resonance appear in FHN neural system under different values of parameters. The effects of the multiplicative noise intensity D and the additive noise intensity Q on the SNR are entirely different. In addition, the discharge behavior of FHN neural system is restrained when the value of Q is smaller. But, it is conducive to enhance signal response of FHN neural system when the values of Q and D are relatively larger.

The transient properties of a bistable system with time-delayed feedback and non-Gaussian noise are investigated. The explicit expressions of the mean first-passage time (MFPT) are obtained. The numerical computations show that the MFPT of the system is affected by the delay time τ, the non-extensive index q and the color noise correlation time τ₀. That is, q can induce the MFPT from a complex behavior to a simple monotonous behavior with τ increasing (i.e. from two extrema to no extremum). But with the q increasing, the MFPT has a extremum, which is more large as τ increases.

We investigate a stochastic model for single species population growth with strong and weak Allee effects subjected to coupling between non-Gaussian and Gaussian colored noise as well as nonzero cross-correlation in between. Stationary probability distribution of population model is obtained depending on the Fokker–Planck equation. The mean first-passage time is also calculated in order to quantify the time of transition between survival state and extinction state with Allee effect in population. The intensity of non-Gaussian colored noise can induce phase transition, and population may be vulnerable to extinction due to the increase in the intensity of non-Gaussian colored noise. Whether Allee effect is strong or weak, the increase in Allee threshold will not contribute to the survival and stability of the population. Further, the phenomenon of resonant activation is firstly discovered in the study of population dynamics with Allee effect. These behaviors can be interpreted on the basis of a biological model of population evolution.

In this paper, the stable state transformation and the effect of the stochastic resonance (SR) for a metapopulation system are investigated, which is disturbed by time delay, the multiplicative non-Gaussian noise, the additive colored Gaussian noise and a multiplicative periodic signal. By use of the fast descent method, the approximation of the unified colored noise and the SR theory, the dynamical behaviors for the steady-state probability function and the SNR are analyzed. It is found that non-Gaussian noise, the colored Gaussian noise and time delay can all reduce the stability of the biological system, and even lead to the population extinction. Inversely, the self-correlation times of two noises can both increase the stability of the population system and be in favor of the population reproduction. As regards the SNR for the metapopulation system induced by the noise terms and time delay, it is discovered that time delay and the correlation time of the multiplicative noise can effectively enhance the SR effect, while the multiplicative noise and the correlation time of the additive noise would all the time suppress the SR phenomena. In addition, the additive noise can effectively motivate the SR effect, but not alter the peak value of the SNR. It is worth noting that the departure parameter from the Gaussian noise plays the diametrical roles in stimulating the SR effect in different cases.

In this paper, the mean first-passage time (MFPT) in simplified FitzHugh–Nagumo (FHN) neural model driven by correlated multiplicative non-Gaussian noise and additive Gaussian white noise is studied. Firstly, using the path integral approach and the unified colored-noise approximation (UCNA), the analytical expression of the stationary probability distribution (SPD) is derived, and the validity of the approximation method employed in the derivation is checked by performing numerical simulation. Secondly, the expression of the MFPT of the system is obtained by applying the definition and the steepest-descent method. Finally, the effects of the multiplicative noise intensity D, the additive noise intensity Q, the noise correlation time $\tau$, the cross-correlation strength $\lambda$ and the non-Gaussian noise deviation parameter q on the MFPT are discussed.

The transport phenomenon (movement and diffusion) of inertia Brownian particles in a periodic potential with non-Gaussian noise is investigated. It is found that proper noise intensity Q will promote particles directional movement (or diffusion), but large Q will inhibit this phenomenon. For large value of Q, the average velocity $〈V〉$ (or the diffusion coefficient D) has a maximum with increasing correlation time $\tau$. But for small value of Q, $〈V〉$ (or D) decreases with increasing $\tau$. In some cases, for the same value of Q and the same value of $\tau$, non-Gaussian noise can induce particles directional movement (or diffusion), but Gaussian colored noise cannot.

The ratchet effect of an overdamped Brownian particle subjected to a spatially symmetric potential driven by a biharmonic temporal force is investigated when a non-Gaussian noise is considered. It is found that the noise intensity and the departure from the Gaussian noise will suppress the ratchet transport effect, while the noise color will strengthen the ratchet transport effect. In the same time, the phase difference mainly changes the direction of the transport. It is also shown that the abundant transport phenomenon is due to the breaking of symmetry for the biharmonic force.

Effects of non-Gaussian α-stable Lévy noise on the Gompertz tumor growth model are quantified by considering the mean exit time and escape probability of the cancer cell density from inside a safe or benign domain. The mean exit time and escape probability problems are formulated in a differential-integral equation with a fractional Laplacian operator. Numerical simulations are conducted to evaluate how the mean exit time and escape probability vary or bifurcates when α changes. Some bifurcation phenomena are observed and their impacts are discussed.

We report measurements and analysis of the voltage noise due the to vortex motion, performed in superconducting Niobium micro-bridges. Noise in such small systems exhibits important changes from the behavior commonly reported in macroscopic samples. In the low biasing current regime, the voltage fluctuations are shown to deviate substantially from the Gaussian behavior which is systematically observed at higher currents in the so called flux-flow regime. The responsibility of the spatial inhomogeneities of the critical current in this deviation from Gaussian behavior is emphasized. We also report on the first investigation of the effect of an artificial pinning array on the voltage noise statistics. It is shown that the fluctuations can lose their stationarity, and exhibit a Lévy flight-like behavior.

In this Letter, we study firing transitions induced by a particular kind of non-Gaussian noise (NGN) and coupling in Newman-Watts small-world neuronal networks. It is found that chaotic bursting can be tamed by the coupling and evolves to regular spiking or bursting behavior as the coupling increases. As the NGN's deviation from Gaussian noise changes, the neurons exhibit firing transitions from irregular spiking to regular bursting, and the number of spikes inside per burst varies with the change of the deviation. These results show that the NGN and the coupling play crucial roles in the firing activity of the neurons, and hence are of great importance to the information processing and transmission in the neuronal networks.

In this Letter, we study the effect of the interaction of external non-Gaussian noise and channel noise on the temporal coherence of the collective intrinsic spiking of an array of bi-directionally coupled stochastic Hodgkin–Huxley (HH) neurons, mainly investigating how the non-Gaussian noise's deviation q from Gaussian distribution affects the spiking coherence and coherence resonance (CR) induced by channel noise and neuron number. It is found that the spiking coherence for small channel noise and the CR induced by channel noise or by neuron number change with the variation of q. As q is increased, the spiking with smaller channel noise becomes more ordered in time, and the CR by channel noise moves to bigger patch sizes. Furthermore, there is CR phenomenon when neuron number is varied, and the CR can occur in smaller channel noise when q is increased. These results show that appropriate external non-Gaussian noise can enhance and optimize the temporal coherence of the collective spiking of the coupled neurons when channel noise is sufficiently small, and can help the collective spiking with smaller channel noise reach the most ordered performance at an optimal neuron number. The mechanism underlying the phenomena is briefly discussed in terms of the property of the non-Gaussian noise. These findings could help to better understand the joint roles of external non-Gaussian noise and channel noise in the collective spiking activity of an array of coupled stochastic neurons.

In this Letter, we study the effect of time-periodic coupling strength (TPCS) on the coherence resonance (CR) of spiking behavior induced by a particular kind of non-Gaussian noise in Newman–Watts networks of Hodgkin–Huxley neurons. It is found that the CR by the non-Gaussian noise can be enhanced by TPCS when TPCS frequency is equal to or multiple of the inverse of the refractory period, and can occur in networks with more random shortcuts for TPCS than for constant coupling strength. Furthermore, the CR by the non-Gaussian noise can occur at smaller TPCS frequency when network randomness increases. These results show that the CR by the non-Gaussian noise can be enhanced by TPCS and can occur in more complex networks in case of TPCS. These findings may help to better understand the joint roles of the non-Gaussian noise and TPCS in the spiking activity of the neuronal networks.

In this paper, we report that for a weak signal buried in the heavy-tailed noise, the bistable system can outperform the matched filter, yielding a higher output signal-to-noise ratio (SNR) or a lower probability of error. Moreover, by adding mutually independent internal noise components to an array of bistable systems, the output SNR or the probability of error can be further improved via the mechanism of stochastic resonance (SR). These comparison results demonstrate the potential capability of bistable systems for detecting weak signals in non-Gaussian noise environments.

In the present paper, the stability of the population system and the phenomena of the stochastic resonance (SR) for a metapopulation system induced by the terms of time delay, the multiplicative non-Gaussian noise, the additive colored Gaussian noise and a multiplicative periodic signal are investigated in detail. By applying the fast descent method, the unified colored noise approximation and the SR theory, the expressions of the steady-state probability function and the SNR are derived. It is shown that multiplicative non-Gaussian noise, the additive Gaussian noise and time delay can all weaken the stability of the population system, and even result in population extinction. Conversely, the two noise correlation times can both strengthen the stability of the biological system and contribute to group survival. In regard to the SNR for the metapopulation system impacted by the noise terms and time delay, it is revealed that the correlation time of the multiplicative noise can improve effectively the SR effect, while time delay would all along restrain the SR phenomena. On the other hand, although the additive noise and its correlation time can stimulate easily the SR effect, they cannot change the maximum of the SNR. In addition, the departure parameter from the Gaussian noise and the multiplicative noise play the opposite roles in motivating the SR effect in different cases.

In the present paper, the stability and the phenomena of stochastic resonance (SR) for a FitzHugh–Nagumo (FHN) system with time delay driven by a multiplicative non-Gaussian noise and an additive Gaussian white noise are investigated. By using the fast descent method, unified colored noise approximation and the two-state theory for the SR, the expressions for the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) are obtained. The research results show that the two noise intensities and time delay can always decrease the probability density at the two stable states and impair the stability of the neural system; while the noise correlation time $\tau$ can increase the probability density around both stable states and consolidate the stability of the neural system. Furthermore, the other noise correlation time ${\tau }_0$ can increase the probability at the resting state, but reduce that around the excited state. With respect to the SNR, it is discovered that the two noise strengths can both weaken the SR effect, while time delay $\alpha$ and the departure parameter $q$ will always amplify the SR phenomenon. Moreover, the noise correlation time ${\tau }_0$ can motivate the SR effect, but not alter the peak value of the SNR. What’s most interesting is that the other noise correlation time $\tau$ can not only stimulate the SR phenomenon, but also results in the occurrence of two resonant peaks, whose heights are simultaneously improved because of the action of $\tau$.

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times $\tau$ and ${\tau }_0$ can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay $\alpha$ and the correlation time $\tau$ of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time ${\tau }_0$ of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

Recent advances in data science are opening up new research fields and broadening the range of applications of stochastic dynamical systems. Considering the complexities in real-world systems (e.g., noisy data sets and high dimensionality) and challenges in mathematical foundation of machine learning, this review presents two perspectives in the interaction between stochastic dynamical systems and data science.

On the one hand, deep learning helps to improve first principle-based methods for stochastic dynamical systems. AI for science, combining machine learning methods with available scientific understanding, is becoming a valuable approach to study stochastic dynamical systems with the help of observation data. On the other hand, a challenge is the theoretical explanations for deep learning. It is crucial to build explainable deep learning structures with the help of stochastic dynamical systems theory in order to demonstrate how and why deep learning works.

In this review, we seek better understanding of the mathematical foundation of the state-of-the-art techniques in data science, with the help of stochastic dynamical systems, and we further apply machine learning tools for studying stochastic dynamical systems. This is achieved through stochastic analysis, algorithm development, and computational implementation. Topics involved with this review include Stochastic Analysis, Dynamical Systems, Inverse Problems, Data Assimilation, Numerical Analysis, Optimization, Nonparametric Statistics, Uncertainty Quantification, Deep Learning, and Deep Reinforcement Learning. Moreover, we emphasize available analytical tools for non-Gaussian fluctuations in scientific and engineering modeling.